We propose a simple real-valued generalization of the well known integer-valued Erdos number as a topological, non-metric measure of the `closeness' felt between two nodes in an undirected, weighted graph. These real-valued Erdos numbers are asymmetric and are able to distinguish between network topologies that standard distance metrics view as identical. We use this measure to study some simple analytically tractable networks, and show the utility of our measure to devise a ratings scheme based on the generalized Erdos number that we deploy on the data from the NetFlix prize, and find a significant improvement in our ratings prediction over a baseline.

background or methods

Distance in graphs

Generalized Erdos Numbers

A set S of a connected graph G of order n is called a double monophonic set of G if for every pair of vertices x, y in G there exist vertices u, v in S such that x, y lie on a u − v monophonic path. The double monophonic number dm(G) of G is the minimum cardinality of a double monophonic set. A double monophonic set S in a connected graph G is called a minimal double monophonic set if no proper subset of S is a double monophonic set of G. The upper double monophonic number of G is the maximum cardinality of a minimal double monophonic set of G, and is denoted by dm⁺(G). Some general properties satisfied by upper double monophonic sets are discussed. It is proved  that for a connected graph G of order n, dm(G) = n if and only if dm⁺(G) = n. It is also proved that dm(G) = n − 1 if and only if dm⁺ (G) = n − 1 for a non-complete graph G of order n with a full degree vertex. For any positive integers 2 ≤ a ≤ b, there exists a connected graph G with dm(G) = a and dm⁺(G) = b.

background

/pdf/upper-double-monophonic-number-of-a-graph-3wl94py2fw.pdf

Upper double monophonic number of a graph.

The Kirchhoff index of a connected graph is the sum of resistance distances between all unordered pairs of vertices in the graph. Its considerable applications are found in a variety of fields. In this paper, we determine the maximum value of Kirchhoff index among the unicyclic graphs with fixed number of vertices and maximum degree, and characterize the corresponding extremal graph.

Maximizing Kirchhoff index of unicyclic graphs with fixed maximum degree

The eccentricity of a vertex $v$ is the maximum distance between $v$ and anyother vertex. A vertex with maximum eccentricity is called a peripheral vertex.The peripheral Wiener index $ PW(G)$ of a graph $G$ is defined as the sum ofthe distances between all pairs of peripheral vertices of $G.$ In this paper, weinitiate the study of the peripheral Wiener index and we investigate its basicproperties. In particular, we determine the peripheral Wiener index of thecartesian product of two graphs and trees.

http://comb-opt.azaruniv.ac.ir/article_13596_983abceb15410e89528e5fcbb919dade.pdf

Peripheral Wiener Index of a Graph

We deﬁne the disorder number of a graph as the maximal length of a walk along the edges of the graph, according to any ordering of its vertices. We then reformulate, from this graph theory point of view, the known results regarding the few special cases that were previously studied, putting them thereby in a unifying context. We study, seemingly for the ﬁrst time, the disorder number of the cycle graph and of the grid graph of arbitrary size.

Distance in graphs

Citations

Generalized Erdos Numbers

Upper double monophonic number of a graph.

Maximizing Kirchhoff index of unicyclic graphs with fixed maximum degree

Peripheral Wiener Index of a Graph

The disorder number of a graph

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